$2$-Restricted Optimal Pebbling Number of Some Graphs
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909448767799296 |
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| author | Dehqan, Juma Gul Alikhani, Saeid Khalafi, Ali Delavar Aghaei, Fatemeh |
| author_facet | Dehqan, Juma Gul Alikhani, Saeid Khalafi, Ali Delavar Aghaei, Fatemeh |
| contents | Let $G=(V,E)$ be a simple graph. A pebbling configuration on $G$ is a function $f:V\rightarrow \mathbb{N}\cup \{0\}$ that assigns a non-negative integer number of pebbles to each vertex. The weight of a configuration $f$ is $w(f)=\sum_{u\in V}f(u)$, the total number of pebbles. A pebbling move consists of removing two pebbles from a vertex $u$ and placing one pebble on an adjacent vertex $v$. A configuration $f$ is a $t$-restricted pebbling configuration ($t$RPC) if no vertex has more than $t$ pebbles. The $t$-restricted optimal pebbling number $π_t^*(G)$ is the minimum weight of a $t$RPC on $G$ that allows any vertex to be reached by a sequence of pebbling moves. The distinguishing number $D(G)$ is the minimum number of colors needed to label the vertices of $G$ such that the only automorphism preserving the coloring is the trivial one (i.e., the identity map). In this paper, we investigate the $2$-restricted optimal pebbling number of trees $T$ with $D(T)=2$ and radius at most $2$ and enumerate their $2$-restricted optimal pebbling configurations. Also we study the $2$-restricted optimal pebbling number of some graphs that are of importance in chemistry such as some alkanes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_03004 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $2$-Restricted Optimal Pebbling Number of Some Graphs Dehqan, Juma Gul Alikhani, Saeid Khalafi, Ali Delavar Aghaei, Fatemeh Combinatorics 05C30, 05C69 Let $G=(V,E)$ be a simple graph. A pebbling configuration on $G$ is a function $f:V\rightarrow \mathbb{N}\cup \{0\}$ that assigns a non-negative integer number of pebbles to each vertex. The weight of a configuration $f$ is $w(f)=\sum_{u\in V}f(u)$, the total number of pebbles. A pebbling move consists of removing two pebbles from a vertex $u$ and placing one pebble on an adjacent vertex $v$. A configuration $f$ is a $t$-restricted pebbling configuration ($t$RPC) if no vertex has more than $t$ pebbles. The $t$-restricted optimal pebbling number $π_t^*(G)$ is the minimum weight of a $t$RPC on $G$ that allows any vertex to be reached by a sequence of pebbling moves. The distinguishing number $D(G)$ is the minimum number of colors needed to label the vertices of $G$ such that the only automorphism preserving the coloring is the trivial one (i.e., the identity map). In this paper, we investigate the $2$-restricted optimal pebbling number of trees $T$ with $D(T)=2$ and radius at most $2$ and enumerate their $2$-restricted optimal pebbling configurations. Also we study the $2$-restricted optimal pebbling number of some graphs that are of importance in chemistry such as some alkanes. |
| title | $2$-Restricted Optimal Pebbling Number of Some Graphs |
| topic | Combinatorics 05C30, 05C69 |
| url | https://arxiv.org/abs/2501.03004 |